FINITE-ELEMENT METHODS FOR NEUTRON-TRANSPORT BASED ON MAXIMUM AND MINIMUM PRINCIPLES FOR DISCONTINUOUS TRIAL FUNCTIONS

Variational principles are given for the first order Boltzmann equation for neutron transport and its equivalent second order forms for even and odd-parity transport. For these extremum principles there are no boundary conditions to be imposed, and across interfaces the trial functions can be discontinuous. Numerical solutions are presented for the even-parity principle. The scale of interface discontinuities is controlled by a penalty parameter chi. As chi is increased for a given basis the degree of satisfaction of the transport equation by the optimised trial function decreases to that for a classical trial function, whereas the interface discontinuities decrease. These trends lend themselves to a simple interpretation in terms of vectors in a Hilbert space. In applications the spatial dependence of a trial function is represented by finite elements, and a spherical harmonic expansion gives the directional dependence of the trial function. The principles are applied in the method of composite solutions in which the order of the spherical harmonic expansion can be varied according to the physical needs of the regions of a system. Large values of chi are used in the method of composite solutions. On the other hand specific values of chi when used with the simplest finite element lead to the well-known central difference equations for neutron transport.